Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-28T13:55:56.357Z Has data issue: false hasContentIssue false
  • English
  • Français

Kähler–Einstein metrics with SU(2) action

Published online by Cambridge University Press:  24 October 2008

Andrew S. Dancer  and
Ian A. B. Strachan
Andrew S. Dancer
Affiliation:
Max-Planck-Institut für Mathematik, Gottfried-Claren-Strasse 26, D53225 Bonn 3, Germany
Ian A. B. Strachan
Affiliation:
Department of Mathematics and Statistics, University of Newcastle, Newcastle-on-Tyne, NEl 7RU
Get access Rights & Permissions [Opens in a new window]

Extract

The aim of this paper is to analyse Riemannian Kähler–Einstein metrics g in real dimension four admitting an isometric action of SU(2) with generically three-dimensional orbits. The Kähler condition means that there is a complex structure I, with respect to which the metric is hermitian, such that the two-form Ωdefined by

is closed. It is well-known that if this condition holds then Ω is in fact covariant constant.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 115 , Issue 3 , May 1994 , pp. 513 - 525
Copyright
Copyright © Cambridge Philosophical Society 1994

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Atiyah, M. F. and Hitchin, N. J.. The geometry and dynamics of magnetic monopoles (Princeton University Press, 1988).Google Scholar
[2]Belinskii, V. A., Gibbons, G. W.. Page, D. N. and Pope, C. N.. Asymptotically Euclidean Bianchi IX metrics in quantum gravity. Phys. Lett 76B (1978), 433–435.Google Scholar
[3]Bergery, L. Berard. Sur des nouvelles variétés riemanniennes d'Einstein. Publications de l'Institut E. Cartan (Nancy) 4 (1982), 1–60.Google Scholar
[4]Besse, A.. Einsteina Manifolds. Ergebnisse der Math. und ihrer Grenzgebiete (Springer-Verlag, 1987).Google Scholar
[5]Bouchiat, C. and Gibbons, G. W.. Non-integrable quantum phase in the evolution of a spin-1 system: a physical consequence of the non-trivial topology of the quantum state-space. J. Phys. France 49 (1988), 187–199.Google Scholar
[6]Futaki, A.. Kahler–Einstein metrics and integral invariants. Springer Lecture Notes in Mathematics 1314 (Springer-Verlag, 1988).Google Scholar
[7]Gibbons, C. V. and Pope, C. N.. The positive action conjecture and asymptotically Euclidean metrics in quantum gravity. Commun. Math. Phys. 66 (1979), 267–290.Google Scholar
[8]Jensen, G. R.. Homogeneous Einstein spaces of dimension 4. J. Differential Geometry 3 (1969), 309–349.Google Scholar
[9]Matsushima, Y.. Sur la structure du groupe d'homéomorphismes analytiques d'une certaine variété kählérienne. Nagoya Math J. 11(1957)145–150.Google Scholar
[10]Myers, S. B.. Riemannian geometry in the large. Duke Math. J. 1 (1935), 39–49.CrossRefGoogle Scholar
[11]Pedersen, H.. Eguchi–Hanson metrics with cosmological constant. Classical and Quantum Gravity 2 (1985), 579–587.Google Scholar
[12]Pedersen, H. and Poon, Y. S.. Kähler surfaces with zero scalar curvature. Classical and Quantum Gravity 7 (1990), 1707–1719.Google Scholar